Some complex change point problems can be reduced to
that of detecting changes in the mean
by adopting a suitable data transformation
where typically, the dimensionality of the transformed data
is larger (sometimes much larger) than that of the original observation sequence.
For example, the problem of detecting changes
in the covariance of \(p\) -dimensional data \(\mathbf {X}_t = (X_{1t}, \ldots , X_{pt})^\top \) 
(under the assumption of a constant mean),
can be transformed to that of detecting changes in the mean of \(p (p+1)/2\) -dimensional data
by considering the vector consisting of cross-products,
i.e. \(\mathbb {X}_t = (X_{it} X_{jt}: \, 1 \le i \le j \le p)^\top \) .
Clearly, the effective dimensionality is quadratic
in the dimension of the original data.
Additionally, the components of \(\mathbb {X}_t\)  are not independent
and the dependence is not negligible.
Such a transformation has been adopted for change point analysis in the covariance structure
of multivariate [1]}, functional [2]}
and high-dimensional [3]} data,
and [4]} propose a similarly motivated wavelet transform
for the segmentation of high-dimensional time series in the second-order structure.
